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Proofs in Indian Mathematics - 03 Mathematics in India: from Vedic Period to Modern Times Prof. M. D. Srinivas Department of Mathematics Indian Institute of Technology – Madras Lecture - 38 Proofs in Indian Mathematics - 03 So, this is the third lecture on proofs in Indian mathematics. So, we should concentrate in the large part of this lecture on the proofs of the results that sort of pertain to calculus. The discoveries are Madhava, so where really one has to take this limit of some quantities per large n. So, how does Yuktibhasa handle these kinds of proofs that we will be discussing in this lecture and in the end, we will have some general comments about Upapatti and proof and some comments on history of mathematics. (Refer Slide Time: 00:20) So, as we have repeatedly said the most detailed exposition of proofs or Upapattis in Indian mathematics is found in the Malayalam text Yuktibhasa or Jyesthadeva. Yuktibhasa sets out or states that its purpose is to give the rationale of the results and procedures presented in Nilakantha’s Tantrasangraha. This book is in Malayalam, so many of these rationales have also been presented mostly in the form of Sanskrit versus by Sankara Variyar (FL) of Nilakantha in two commentaries Kriyakramakari on Lilavati and Yuktidipika on Tantrasangraha. (Refer Slide Time: 01:00) Yuktibhasa is 15 chapters. Its chapter 6 which deals with the Madhava series for pi is the chapter on the Paridhi-Vyasa sambandha. Chapter 7 is the chapter on Jyanayana which deals with Madhava series for the Rsine and Rversine. So, let us straight away look at how Yuktibhasa tries to estimate this sum, this is called Samaghata-Sankalita, sankalita is the sum, ghata is a product, Samaghata means equal powers, so sum of natural integers of the same powers. (Refer Slide Time: 02:06) We have to find out how this unbehaves for large n, we have to asymptotic estimate of this sum and this is at the heart of development of calculus, estimating this was a major effort in whole of 17th century efforts in Europe, which arrived ultimately at the development of calculus. Yuktibhasa has a statement of the result and a proof of it, so we shall discuss the proof. First, it is noted that the sum of the first product, we have Aryabhatta’s exact results, N * n+1/2. And obviously for large n, you can neglect the n/2 term and you can say this goes like n square/2. So, it is this capable separation of the orders, in which has n becomes large. What is the significant term or in 1/n has n becomes large, what are the terms that can be neglected, this is where the heart of this limiting operation talks. Now, we are saying to this varga- sankalita. Of course, we know the exact formula is available. So, we can again look at it and write down the asymptotic behavior for large n, but Yuktibhasa choose us to start writing down the proof in the way it will do for the general case, so that you understand the method of proofing the general case. So, it says write this Sn to in this way, n square + n - 1 square, etc + 1 square. Subtract from Sn of 2, n times Sn of 1. N times the mula-sankalita, 1 + 2 + etc. n. And when you subtract nSn1 – Sn2 becomes 1*n-1 + 2*n-2 + 3*n-3 + etc., + n-1*1. Now, remember Yuktibhasa does not have any symbols, any diagrams, all these are set out in terms of sentences, all these equations are explained in terms of sentences, all diagrams are explained in terms of sentences, but to write diagrams, they will tell you start from the east corner, go to the north-west corner, then look at the place where this line cuts the circle. So, they may very nice way of describing the geometrical diagrams. In the same way, this algebraic results are stated in verbal form, so n times Sn-1 – Sn2, therefore can be written as n-1, then these 2 n-2 can be written this way, the 3 n-3 can be written that way, the n-1 ones can also, so now you re-sum them that is now you look at this sum this way, you look this sum this way. So, this is mula-sankalita up to n-1, this is mula-sankalita up to n-2 etc. So what we have, so nSn1 – Sn of 2 is Sn-1 1, Sn-2 1, Sn-3 1 that is mula-sankalita going up to n-1, mula-sankalita going up to n-2, so this is the, not here, we can look at it once again Sn2 - nSn1, we write like this and then rewrite it and sum it horizontally now and therefore, we have this ending. So this is at the basis of the Yuktibhasa proof actually. Later on, generalize this identity and prove the result by induction. (Refer Slide Time: 05:16) So, since we have already estimated the mula-sankalita summations of the first powers of natural integers go like n square by 2, so we substitute here, so Sn-1 we write it as n-1 whole square by 2, Sn-2 1 is written as this, Sn-3 is approximated by this and so on and now, we again behold that this is sum of the squares of integers, of course for large n, the later terms are not of significant and so this whole sum can be written as the square of the first n-1 integer. The sum of the squares of first n-1 integers divided by 2. So, you have an equation like this. Now, this Sn-1 of 2 and Sn of 2 are almost the same. When n is very large, there is no difference between the sum of the squares of the first n integers and the sum of the squares of first n-1 integers, at least for the purpose of the large n behavior, which we are estimating now. So, these two can be taken to be the same. Therefore, taking this to the right hand side, this will become Sn of 2, 3/2 of it and here already nSn1 is there, therefore finally you will get Sn of 2 goes like n cube - 3, in case this step is not clear, what we have shown is nSn of 1 - Sn of 2 goes like Sn-1 of 2/2, right, this what we have shown. Now, you take this to the right hand side, so you just get nSn of 1 approximately = Sn of 2, 3 times, right. These two are the same and this has already been estimated. (Refer Slide Time: 07:16) So, Sn of 2 goes like 2n/3 Sn of 1 what is Sn of 1, it goes like n square/2, we have already estimated that for large n, therefore we have Sn of 2 goes like n cube by 3. All this is set out verbally in a Yuktibhasa or in Kriyakramakari commentary on Lilavati. Similarly, it is shown that the sum of the third powers of integers goes like n to the power 4/3 for large n. This argument is also explained in detail in Yuktibhasa, same kind of argument. (Refer Slide Time: 08:21) Now follows the argument by mathematical or using mathematical induction to demonstrate the same estimate for a general Samaghata-Sankalita. So, he is saying suppose you are interested in summing the Samaghatas where it goes up to (FL) some number. For that, you take first n times the Samaghata-Sankalita the lower order and subtract the Samaghata- Sankalita of the given order. So, same thing that we did for nSn1 - Sn2, then this argument is given by recombining these terms that this is actually = the repeated summation of Sn-1 of k-1, the varasankalita of the lower order Samaghata-Sankalita that how it is explained. So, this relation is just like this relation. Only this is k-1 and this is k, the argument is very similar to prove this. So, this is the basic relation and this relation there is no estimate involved, it is exact. Now, we argue what happens when n becomes large. So, in all of Yuktibhasa up to a point, the terms of all n are kept and its exact results are written and then the final limit and arguments are made. So, this result is exact that nSn k-1 - Sn of k is this. Now, if we have already estimated that is how a mathematical induction argument works. First you write down a relation, then assume something for n and then show it for the n+1 and then you have shown it for all n, if you have already shown it for n = 1. (Refer Slide Time: 09:59) So, assume that the k-1 called as Samaghata-Sankalita has already been estimated and it goes like n to the power k/k. Once you assume that and put that into this equation, so this has already been estimated. So each of this go like n-1 to the power k/k-1, n-2 to the power k/k, each of them goes like this, n-1 to the power k/k, n-2 to the power k/k, n-3 to the power k/k. So, this is looking of course terms which are very small in the other end do not matter.
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